When Arthur F. Burns, chairman of the Federal Reserve Board, was at the height of his popularity last spring, the betting was he might be in office for another 4 1/2 years. Although Burns is a Republican, and 73, policy watchers were speculating that President Carter would reppoint him when his term runs out this January, if only as a gesture to bolster business confidence. To many observers, it seemed that Arthur Burns could do no wrong.

Today, just a few months after the economy's heady first quarter, the odds have shifted dramatically - and not in favor of the Fed chairman's tenure. Burns and the administration are looked in a swrious dispute over monetary policy. Democrats are criticizing him for intransigence, and top Cater advisers are dead set against his reappointment. "The odds are virtually nil for Arthur's getting a third four-year term," one bet-taker here asserts.

There's no real arguing that Burns' prospects have dimmed visibly in recent months. The most telling signs may be his own behavior: Burns himself has been acting like a man who knows his days in office are numbered. Several times in the past few months, he's publicly rebuffed White House appeals to end the Fed's recent credit-tightening. And only last week, he delivered a major attack on the administration's overall economic policy.

Yet, the question of Burns' reappointment - or who will succeed him - continues to be one of the most titillating in Washington. Businessmen regularly warm against dropping him. Economists argue over the decision's impact on the recovery. And liberals spend their lunch hours drawing up lists of replacements. "Arthur is having almost as much impact in stirring up speculation about his reppointment as he does on the money supply," jokes one local wag. "I only hope it has a better result."

To be sure, the Burns nomination is no routine reappointment. The decision Carter makes next January will have a major impact in determining the nation's money and credit policies - and possibly the shape of the recovery - for many years to come. "It's a lot more crucial than, let's say, who is to head the Office of Management and Budget," says Henry Kaufman, partner in Salomon Brothers, the New York investment house. "This really is a policy move."

The question of Burns' tenure isn't a new one with the administration. As the chairman's defenders point out with relish, when Carter first took office, it was virtually assumed he would replace Burns when he got the opportunity. The former Georgia governor had made the Fed chairman a frequent target during the campaign. And there was strong pressure from blacks and labor to put a moderate Democrat at the monetary helm.

But as the economy immproved, Carter seemed to warm to his old economic nemesis, taking him into confidence on the administration's budding stimulus program and eventually praising him publicly. Last May, when the two received honorary doctorates from Notre Dame University, Carter conspicuously walked across the platform to congratulate Burns. And late last summer, the two began monthly lucheon-dates - a virtually unprecedented move.

Moreover, Burns seemed to enjoy unchallenged prestige both in Congress and in the business community. Apart from his legendary skill as a politician, the chairman had been proved right on his newly held theory that the economy no longer needed as rapid a rate of growth in the money supply. And conservatives saw the Fed as their champion against inflation. Says one onlooker, "The odds against his reappointment dropped from 50 to 1 to 2 to 1.

But this unlikely alliance has been ended abruptly both by the credit-tightening that the Fed began in earnest this past fall - a move that has driven interest rates up sharply, and has worried top Carter policy makers - and by the increasing reality of Burns outspoken independence. The Fed chairman's opposition played a major role in killing the President's proposal last spring for a $50-a-person tax rabate. And Burns has opposed a key Carter move to end the tax break for capital gains.

Indeed, to many observers, Burns' own rhetoric seems to be more painful to the administration than the Fed's credit-tightening. While top White House officials refuse flatly to discuss the flap, others familiar with the dispute say policy makers are concerned that Burns may have talked himself into a box by emphasizing money supply growth as the Fed's principal guidepost - a situation some fear constrains the administration in handing budget and tax policy.

They're more worried about the future than they are about the present," says one knowledgeable source here. Burns has been so rigid that the market now reacts adversely no matter what the Fed does with the money supply. If monetary growth spurts, the stock market slumps because it fears inflation will result. But if the Fed tightens any, investors get scared because interest rates are going up. You just can't win. There's no more room to move further."

Officials also reportedly are one edge over what they see as Burns' reluctance to act as a team member. One source says that when the Fed chairman meets with top economic policy makers, he "just lectures them. There's no opportunity for give and take." And although Carter and Burns are said to get along well enough in personal meetings, the chairman has told friends that he still doesn't "know what the President stands for, and I don't think anybody does."

Worse, from the administration's point of view, the indomitable Burns has't budged an inch to accommodate the new President, either in relaxing money and credit policies or in easing up on his criticisms. "I see nothing in Arthur's behavior over the past 10 months that shows he's move even a smidgeon in either direction," mutters one outsider close to the administration. "Why should the President put up with it?" he asks. (Burns declined a request by The Post for an no-the-recod interview.)

Nor is this the first time that Democrats have been dissatisfied with Burns' policies. The Fed chairman drew fire in 1972 (unjustly, some believe) on charges of having bloated the money supply to help re-elect Richard M. Nixon - a move critics say contributed to the 1973 price spurt. And the Burns-led board generally is blamed for having overreacted in late 1975, when it reined in on money and credit policies after panicking over the impact of the Ford tax rebate. Burns since has admitted the agency has made a "mistake."

Critics also complain about occasional flashed of Burnsian "arrogance" - a trait that ires some junior congressmen and sensators, though veterans say it is more muted now than it was five or six years ago. Still, Burns is credited with uncanny skill as a politician. Both at public hearings in Conngress and in private sessions with lawmakers, he's proved adept at winning support even from his detractors.

The problem is, for all the administration's frustrations over Burns' recent performance, officials conncede it won't be that easy either to get rid of him or replace him. For one thing, the silver-haired economist has become a symbol to conservatives - the lone man in the government, they say, who still cares seriously about fighting inflation. Some argue that to dump Burns now would risk heightening inflationary psychology.

For another, some strategists point out that Carter may well find in useful to have a conservative at the belm of the Fed - it only to serve as a political lightning rod. The argument is that keeping Burns in office could provide Carter with the over he needs to pursue a lesstringent fiscal policy. And, as Nixon showed when he broke the ice with China, a conservative sometimes can get away with more liberal policies than a liberal - if he's willing to do it.

(The impact of the reappointment question on the markets was illustrated graphically earlier this fall when a news service ran reports of rumors that Burns had decided to resign. Although the reports proved erroneous - and quoted sources in Paris, not the U.S. - both stock prices and the value of the dollar declined. Salomon Brothers' Kaufman asserts reassuringly that the markets "would go on anyway" if the right successor were chosen. But not all conservatives seem convinced.)

The main question is, whom should Carter pick to replace Burns if he decides to dump the Fed chief? Although the President has given no hint he's even thought about the prospect, there is a standard list: At the top are Paul A. Volcker, president of the New York Federal Reserve Bank (and former undersecretary of the Treasury for monetary affairs), and Robert V. Roosa, who held the same Treasury post in the Johnson administration.

Others cited routinely by analysts include Arthur M. Okun, Johnson's chief economic adviser; Andrew F. Brimmer, former Federal Reserve Board member, Gabriel Hauge, chairman of Manufacturers' Hanover Trust Co.; David L. Grove, vice president and chief economist of International Business Machines Corp.; and Bruce K. MacLaury, former president of the Minneapolis Federal Reserve Bank, and chairman of the Brookings Institution. Hauge said last week he would not accept the post.

Okun already has taken himself out of the running, on grounds that his image is "too liberal" to satisfy the business community. And most outside observers place Brimmer in that same category. "The markers would have a fit if a liberal were appointed," concedes one key official involved in the fracas. "If Carter wants to depress business confidence for the next umpteen months, that's all he's got to do."

Kaufman holds similar views. "The President really has one basic choice," he says. "He either can rreappoint Dr. Burns or else he can bring in someone else with a somewhat comparable persuasion. But whomever he appoints has to be someone strongly identified with a stiff anti-inflation posture. And he's got to be representative in a moralistic sense of the integrity of the dollar. You can appoint a liberal during a recession, but it won't work right now."

Monetary quarterbacks also advise against choosing a chairman with rigid ideological leanings - either towards the monetarist school, which emphasizes a steady, moderate growth in the money supply, or the Kenesian branch, which places more stress on behavior of interest rates. (Insiders say Carter is likely to choose neither if he decides to replace Burns, if only because he himself is such a pragmatist.)

With those qualifications, observers say Volcker and Roosa would seem to make the most likely candidates. Volcker, in particular, shares much in common with the present chairman - a reputation as an inflation fighter, respect of the international community, broad experience in monetary matters. "He's a slightly more liberal version of Burns," one onlooker muses. "Enough probably to satisfy the administration, but not to make outsiders nervous."

Ironically, however, Volcker's appointment - and those of other potential candidates - could be blocked by a technical difficulty: Under present law, the Fed's seven govenors must be chosen from among different districts, or sections of the country - and Burns has the New York seat tied up. It's only Burns' term as chairman that expires next Jan. 31 - and, if the chooses, he may stay on the board until 1984.

If that happens, Volcker, Roosa, Grove andd Hauge all would be barred from running. And the continuation of recently appointed J. Charles Partee in the Richmond seat would rule out Washingtonians Okun, Brimmer and MacLaury. The Fed chairman then would have to come from a so-called "vacant" district. The only ones remaining would be Minneapolias, San Francisco, Kansas City, Chicago, St. Louis or Cleveland. And that means a virtual unknown.

The effect of the district-vacancy hitch would be effectively to give Burns a veto, if he wants to use it, on the man Carter names to succeed him - a threat that worries some officials. But Kaufman, for one, thinks such fretting most likely is for naught. "It just wouldn't be typical of Arthur Burns to stay on the board if he's not reappointed as chairman," he says. "It's not a very realistic option."

At the same time, however, Kaufman and others say they can well envision Burns taking such a move if he harbors "serious reservations" about the man Carter has named as his replacement. "It's an outside possibility," one Fed-watcher says. Most analysts doubt strongly that Burns would move to block Volcker, in particular. "They think too much alike," a Wall Streeter says.

At present, almost all sides concede that Burns' chances for reappointment look decidedly bleak. Carter's top two economic policy makers, Charles L. Schultze, chairman of the Council of Economic Advisers, and W. Michael Blumenthal, the Secretary of the Treasury, both are said to oppose his renomination, and a third - Stuart E. Eizenstat, the President's chief domestic adviser - has been a Burns detractor from the start.

And Democrats outside the White House are clamoring for a more liberal replacement. Rep. Henry S. Reuss (D-Wis.), the liberal chairman of the House Banking Committee and a sometime defender of Burns, says, a turnover is necessary. "Without any denigration of Dr. Burns, the times call for something new," he said in an interview. (Reuss proposes liberal economist James Tobin of Yale University, but insiders say that appointment is unlikely.)

Just the same, no one is counting Burns out yet. if only because he's such a master politician. "Arthur got Congress to accept his theory on the money supply and to kill off the rebate," a Burns admirer notes gleefully. "Maybe - just maybe - he can pull this one off as well."

Indeed, conservatives already have begun a major campaign to prod the White House into keeping the Fed chairman. Officials report that every time they talk with businessmen these days, the company executives invariably lobby for Burns' reappointment.And key congressmen report they, too, have been under pressure to push for a third term for Burns.

In fact, Burns himself may not quite have given up. His artfully drafted speech last week was, on one hand, an indictment of the administration. But some observers believe it also was cleverly designed to rally support from business, which looks to the chairman to bolster its confidence. (A few days after the Burns talk, Schultze embraced Burns' calls for hefty business tax cuts, and dropped any mention of monetary policy.)

And the chairman has stepped up his "visits" to regional Federal Reserve Banks - in what some observers interpret as a lobbying move. This Wednesday, he's scheduled to meet with the board of the Cleveland bank, in a closed-door session. "Whenver he comes to town," one outlander suggests stoically, "he dispenses prestige upon us all."

Some Burnsophiles here believe that what Carter finally decides will depend in part on the state of the economy at year's end. If the recovery appears to be reasonably strong and inflation seems likely not to accelerate, monetary policy would be less of an issue, and the administration itself would be under less pressure. And some analysts believe tha interest-rate problem may well resolve itself by then, as market conditions come more into balance.

If Carter does decide to replace Burns, it would mark the first time a Fed chairman effectively was deposed since Marriner S. Eccles was rebuffed by President Truman in 1948. (Burns has not said yet what he would do if reappointment did not come, but friends say he may take a post at the Hoover Institute, the conservative think tank at Stanford University.)

But it also would be the end of an era. Burns, who has served as Fed chief since February 1969, has been a force unto himself at the Fed, both in altering its approach and policy and in building a strong image.